Problem 24 - Thinking Mathematically

Question

Complete the truth table for the given statement by filling in the required columns. $$ \begin{aligned} &p \wedge \sim q\\\ &\begin{array}{|cc|c|c|} \hline \boldsymbol{p} & \boldsymbol{q} & \sim \boldsymbol{q} & \boldsymbol{p} \wedge \sim \boldsymbol{q} \\ \hline \mathrm{T} & \mathrm{T} & & \\ \hline \mathrm{T} & \mathrm{F} & & \\ \hline \mathrm{F} & \mathrm{T} & & \\ \hline \mathrm{F} & \mathrm{F} & & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The truth table completed:\[\begin{array}{|cc|c|c|}\hline p & q & \sim q & p \wedge \sim q \\hline T & T & F & F \\hline T & F & T & T \\hline F & T & F & F \\hline F & F & T & F \\hline\end{array}\]

Step 1: Find the value of \(\sim q\)

Given values of q, the value of \(\sim q\) is just the inverse of q. Therefore, for q=T, \(\sim q\)=F and for q=F, \(\sim q\)=T. Substitute these values in the column for \(\sim q\).

Step 2: Find the value of \(p \wedge \sim q\)

The output of AND gate (\(\wedge\)) is true only if both inputs are true. Taking the values of p and \(\sim q\), if both are true, \(p \wedge \sim q\)=T. For all other combinations, \(p \wedge \sim q\)=F. Substitute these values in the column for \(p \wedge \sim q\).

Key Concepts

Logical Operators

Logical operators are fundamental to understanding how statements can be evaluated in logic and computer science. There are several types of logical operators, but the most common ones are AND (conjunction), OR (disjunction), and NOT (negation). These operators are used to combine or modify logical statements and determine the truth value of more complex expressions based on the truth values of their components.

Logical operators help in formulating truth tables, which are a vital tool in logic to systematically evaluate how the truth value of compound statements is derived from its simpler sub-statements. By applying logical operators, we can determine various outcomes of logical expressions depending on given input values.

Logical Negation

Logical negation is one of the simplest, yet most powerful tools in logic. It is used to reverse the truth value of any given statement. This is commonly represented by the symbol \( \sim \), and it essentially turns a true statement into a false one, and vice versa.

For example, if we have a statement \( q \) that is true, applying logical negation \( \sim q \) will make the statement false. Conversely, if \( q \) is false, then \( \sim q \) will be true. Negation is crucial for expressing logical complements, which are important in formulating complex logical expressions and determining overall truth in systems like the truth table featured in the exercise.

• If \( q = T \), then \( \sim q = F \)
• If \( q = F \), then \( \sim q = T \)

Logical Conjunction

Logical conjunction is the operation that combines two logical statements where both must be true for the collective expression to be true. This is typically represented with the symbol \( \wedge \) (AND). It is essential in creating conditions where multiple criteria must be satisfied simultaneously. In the context of programming and formal logic, it ensures that all specified conditions must hold true.

In a truth table, logical conjunction produces a true output only when both inputs are true. For instance, if we consider two statements \( p \) and \( \sim q \), the expression \( p \wedge \sim q \) will evaluate to true only if both \( p = T \) and \( \sim q = T \). In all other scenarios, the result is false, which is reflected in the corresponding column of the truth table.

• \( T \wedge T = T \)
• \( T \wedge F = F \)
• \( F \wedge T = F \)
• \( F \wedge F = F \)